Multidimensional Approach in Math Education

 

 

Carol Dweck herself has written that the information on the value of changing mindsets needs to be accompanied by a different approach to teaching, one that enables students to learn differently. One of the things that she says keeps her up at night is when students are told to put in effort and that success is all about hard work, without their being given the tools by teachers to learn more effectively. As she says, “Effort is key for students’ achievement, but it is not the only thing. Students need to try new strategies and seek input from others when they are stuck.”

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Fluid and flexible brains, neuroscientists conclude, come from the synchrony that occurs when multiple brain areas are working together. Communication between brain areas comes about when we approach knowledge through multiple avenues, encountering ideas in different forms and representations.

A multidimensional approach can be used in the teaching of all subjects to bring about higher engagement and achievement. Many subject areas, particularly in the humanities, already value treating the subject in multiple ways by asking students to give their own interpretations of texts they read and employing such forms as group discussions, debates, and plays. In most cases they could still become more multidimensional, but they are rarely as narrowly taught as some other subjects. In my experience the subjects that seem most in need of change are mathematics, science, and language teaching. Coming at the subject matter from multiple angles is an ideal learning approach for all of these disciplines.

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Working in these multiple ways encourages brain communication while also bringing the content to life. The vast majority of students think about math as a set of numbers and methods and about English as books and words. When we approach math, English, science, or other subjects as opportunities for creativity and seeing things in multiple ways, it changes everything, stimulating vital brain growth and neural connections. Additionally, as teachers diversify the curriculum, moving from a simple list of numerical answers, pages of text, or scientific equations to visuals, models, words, videos, music, data, and drawings, the classroom changes from a place where all the work looks the same to one where the variety is enticing and creativity can be celebrated.

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This new openness to challenges and uncertainty seems to be a common reaction to becoming unlocked—people realize that it is good to struggle, that it is not a sign of a brain weakness, but of brain growth. This leads to more confidence in times of struggle and a willingness to share ideas that they are unsure of. One of the saddest, most central characteristics of fixed-brain thinking is the fear of being wrong. People’s minds are literally locked, immobilized, by their fear, which is why an approach to life that values multidimensionality, growth, and struggle is so liberating. Holly said: “I have so many more ideas because I let myself have ideas.”

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Another core benefit of working and living with a multidimensional approach is that when roadblocks appear, you know there are alternate routes. Many of the adults I interviewed for this book said they would no longer stop when they met challenges or roadblocks; they simply would find another strategy, another approach. A multidimensional approach to knowledge reveals that there isn’t only one way to do anything; there are always multiple ways forward.

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In many classrooms students are given problems they do not know how to access—which causes them to think negatively about themselves and their learning. When problems are changed to become “low floor and high ceiling”—problems that are accessible by all but lead to more challenging work—everybody can access them and take them to different places.

You do not have to live your life as an “expert”

 

Part of the process of change and of becoming limitless involves letting go of the idea that your past failures came about because there was something wrong with you. A similarly important change is realizing that you do not have to live your life as an “expert,” that you can go into situations and proudly share uncertainty. Jesse Melgares told me about these two aspects of the change he went through as he became unlocked. Jesse is an assistant principal in east LA, but in earlier years he taught mathematics and was, as he said, “extremely self-conscious,” thinking he did not know enough and nothing could change. When Jesse became an assistant principal, he needed to coach math teachers, but he was fearful that others would find out he was a fraud:

To be honest I would get a lot of paralyzing stress when someone asked me a math-related question. . . . It was terrible. It was like a boot on my chest. It’s what I woke up with in the morning, wondering, “Am I gonna be asked something that I don’t know the answer to? And will I be discovered as some sort of fraud?”

The feeling of paralyzing stress Jesse described, the fear of being asked something he could not answer, is a feeling shared by millions of people in different situations and jobs, and it is a feeling that I hope this book can change. For Jesse the change began when he took one of my online courses and realized: “Everything that I had been taught as a student of math when I was in the K–12 system and as a math educator was wrong.”

For Jesse, the first step in becoming unlocked was realizing that any trouble he had had learning in the past was not due to some deficit in him, but to the faulty system in place. This is a shift I have seen others make, and it is vital for those who have had bad learning experiences.

Jesse not only started feeling better about mathematics; he started a new “journey” discovering that mathematics was his passion. He shifted from feeling defeatist about math to seeing it as an exciting challenge. Jesse is now the director of mathematics for twenty-five schools—quite a change for a person who used to feel paralyzing stress when he thought about math. New knowledge about the brain allowed him to shift his perspective, his mindset, and his belief in himself. Jesse still meets questions he cannot answer, but instead of being afraid, he thinks: “Well, I don’t know what the answers are but, you know, we’ll figure it out. This is a challenge.” This shift in perspective is typical for people who have become unlocked. When people change their mindset and become aware of the positive benefits of struggle, they take a new and much more positive approach to challenge and uncertainty. They let go of the need to be the expert and replace it with curiosity and the desire to collaborate.

Beliefs vs Brains

 

In order to study the impact of our beliefs on our health, Stanford researchers Alia Crum and Octavia Zahrt collected data from 61,141 people over an extensive time span, twenty-one years. The researchers found that those people who thought they were doing more exercise were actually healthier than those who thought they were doing less, even when the amount of exercise they were doing was the same.

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If you enter a challenging situation believing in yourself, but then mess up, your brain will react more positively than if you go into a situation thinking, “I don’t think I can do this.” If we have a difficult job or a problematic situation at home, this result should prompt us to go into those situations believing in ourselves. If we enter difficult situations with positive beliefs, our brains will become more resilient and adaptative when we make errors than if we are doubting ourselves. This change in belief alters the physical structures of the brain and creates avenues for higher-level thinking and creative problem solving. Just as those who believed they were engaging in healthy exercise became healthier, those who believe they are learning more productively actually learn more.

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One of Dweck’s studies revealed the immediate impact of the word “smart.” Two groups of students were given a challenging task. On completion, one group was praised for being “really smart,” and the other was praised for working hard. Both groups were then offered a choice between two follow-up tasks, one that was described as easy and one that was described as challenging. Ninety percent of the students praised for working hard chose the harder task, whereas the majority of the students praised for being “smart” chose the easy task. When students are praised for being smart, they want to keep the label; they choose an easy follow-up task, so they can continue to look “smart.”

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It is just as important to take on ideas about social comparison with students as it is to make them aware of the value of struggle. I have had many conversations with learners of all ages who argue that brains must be fixed, because some people appear to get ideas faster and to be naturally “gifted” at certain subject areas. What they do not realize is that brains are growing and changing every day. Every moment is an opportunity for brain growth and development. Some have simply developed stronger pathways on a different time line. It is critical that students understand that they too can develop those pathways at any time—they can catch up with other students if they take the right approach to learning.

Making mistakes are the best times for brain growth

Most of us have grown up with the idea that mistakes are bad, especially if we attended test-driven schools, where we were frequently marked down for making mistakes, or our parents punished mistakes with harsh words and actions. This is unfortunate, and this is why. 

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... [In a workshop, Carol Dweck] announced that every time we make mistakes, synapses fire in the brain, indicating brain growth. All the teachers in the room were shocked, as they had all been working under the premise that mistakes are to be avoided. Carol was drawing from work that has researched the brain’s response when we make mistakes, particularly investigating the different ways brains respond when people have a growth or a fixed mindset.

Jason Moser and his colleagues extended Carol’s work investigating the brain’s response when we make mistakes. Moser and his team found something stunning. They had asked participants to take tests while they monitored the participants’ brains with MRI technology. They looked at the scans when people got questions correct and when they got them incorrect. The researchers found that when people made mistakes, brains were more active, producing strengthening and growth, than when people got work correct. Neuroscientists now agree that mistakes positively contribute to the strengthening of neural pathways.

This learning key is particularly significant because most teachers design classes so that everyone is successful. Curricula and textbooks are designed with trivial, unchallenging questions, so that students will get a high percentage of answers correct. The common belief is that getting most answers correct will motivate students toward greater 

success. Here’s the problem, though. Getting questions right is not a good brain exercise.

For students to experience growth, they need to be working on questions that challenge them, questions that are at the edge of their understanding. And they need to be working on them in an environment that encourages mistakes and makes students aware of the benefits of mistakes. This point is critical. Not only should the work be challenging to foster mistakes; the environment must also be encouraging, so that the students do not experience challenge or struggle as a deterrent. Both components need to work together. 

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One of the significant characteristics of the highly effective learning described is the presence of mistakes and the role of struggle and error in transforming people from beginners into experts. This is consistent with the brain research showing increased brain activity when people struggle and make mistakes and decreased activity when they get work correct. Unfortunately, most learners think they should always be getting work correct, and many feel that if they make mistakes or struggle, they are not good learners—when this is the very best thing they can be doing.

Practice is important for the development of any knowledge or skill. Anders Ericsson helped the world understand the nature of expert performance and found that most world-class experts—pianists, chess players, novelists, athletes—practiced for around ten thousand hours over twenty years. He also found that their success was not related to tests of intelligence but to the amount of “deliberate practice” they undertook. Importantly, although people succeed because they are trying hard, the people who become experts are trying hard in the right way. A range of different researchers describe effective practice in the same way—people pushing at the edge of their understanding, making mistakes, correcting them, and making more.

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Japan has always scored well in mathematics—it has always finished in one of the top-five TIMSS positions—and was one of the countries visited in the study. The researchers found that Japanese students spent 44 percent of their time “inventing, thinking, and struggling with underlying concepts,” whereas students in the US engaged in this kind of behavior less than 1 percent of the time.

Jim Stigler, one of the authors of the study, writes that the Japanese teachers want the students to struggle—and recalls the times when they would purposely give the wrong answer so that students would go back and work with foundational concepts. In my thousands of observations of classrooms over many years in the US and the UK, I have never seen this kind of practice; more typically I have seen teachers who seem to want to save students from struggle. Many times I have observed students asking for help and teachers structuring the work for students, breaking down questions and converting them into small easy steps. In doing so they empty the work of challenge and opportunities for struggle. Students complete the work and feel good, but often learn little.

I saw a very similar teaching approach, focused on struggle, in a visit to classrooms in China, another country that scores highly in mathematics. I had been asked to visit China to give a talk at a conference and managed, as I like to do, to sneak away and visit some classrooms. In a number of high-school math classrooms, lessons were approximately one hour long, but at no time did I see students working on more than three questions in one hour. This contrasts strongly with a typical US high-school math classroom, where students chug through about thirty questions in an hour—about ten times more. The questions worked on in Chinese classrooms were deeper and more involved than the ones in US classrooms. Teachers would ask provocative questions, deliberately making incorrect statements that students would be challenged to argue against.

One of the lessons I watched was on a topic that is often uninspiring in US classrooms—complementary and supplementary angles. The teacher in China asked the students to define a complementary angle, and the students gave their own ideas for a definition. Often the teacher would push the students’ definition to a place that made it incorrect and playfully ask, “Is this right, then?” The students would groan and try to make the definition more correct. The teacher bantered with the students, playfully extending and sometimes twisting their ideas to push the students to deeper thinking. The students probed, extended, clarified, and justified for a long time, reaching depths that were impressive.

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"I am going to figure this out if it kills me."

When I talk with teachers, they often say this sort of persistence is missing in the students they teach. One of the most common complaints I hear from teachers is that students don’t want to struggle; they want to be told what to do. To the teachers it seems as though students just can’t be bothered with struggling, which is probably what it looks like. The truth is, however, that when students don’t want to struggle, it is because they have a fixed mindset; at some point in their lives they have been given the idea that they cannot be successful and that struggle is an indication that they are not doing well.

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As an academic, I experience a lot of failure. To keep our youcubed center at Stanford running, supporting staff salaries and providing free materials for teachers and parents, we have to apply for lots of grants—most of which are rejected. I also have to submit our papers to journals, where rejection is part of the process. If they are not rejected, they are subject to reviewers’ comments. I have had reviewers dismiss my work entirely, saying that it is “not research, just a story.” It is nearly impossible to keep going as an academic without viewing “failure” as an opportunity to improve. A wise professor named Paul Black, my PhD advisor, once said to me: “Whenever you send a paper to a journal, have in mind the next journal you will send it to when the paper is rejected.” I have used his advice a number of times.

Taking a limitless approach—particularly when embracing challenge and struggle—also helps when we encounter difficult people. In today’s world of social media, it seems impossible to make a statement about anything without getting pushback, some of it aggressive. I have experienced extreme and aggressive pushback many times, and I now know that it is important to stay strong in those moments and to look for something positive. Instead of dismissing a challenge or beating yourself up, think, “I will take something from this situation and use it to improve.”

Your brain is constantly reorganizing, growing, and changing

The first step in living a limitless, unlocked life is to know brains are constantly reorganizing, growing, and changing. Remembering that every day of our lives, we wake up with a changed brain. In every moment of our lives our brains have opportunities to make connections, to strengthen pathways, and to form new pathways. When we face a challenging situation, rather than turn away because of fear of not being good enough, we should dive in, knowing that the situation presents opportunities for brain growth. As we start to recognize the huge implications of the adaptability of our brains, we will start to open our minds, and live differently.

Giftedness??

 

 

The idea of giftedness is not only inaccurate and damaging; it is gender and racially biased. We have many different forms of evidence showing that those who believe in fixed brains and giftedness also believe that boys, men, and certain racial groups are gifted and girls, women, and other racial groups are not.

One of the forms of evidence that shows this clearly was collected by Seth Stephens-Davidowitz, who focused his attention on google searches. His study revealed something very interesting and disturbing. He found that the most commonly googled word following “Is my two-year-old son . . .” is “gifted.” He also found that parents search the words “Is my son gifted?” two and a half times more than the words “Is my daughter gifted?” This is despite the fact that young children of different genders have equal potential.

Sadly, the problem is not limited to parents. Daniel Storage and his colleagues conducted analyses of anonymous reviews on RateMyProfessors.com, and they found that students were twice as likely to call male rather than female professors “brilliant” and three times as likely to call male rather than female professors “geniuses.”20 These and other studies show that ideas of giftedness and genius are intertwined with racist and sexist assumptions.

I am convinced that the majority of people who have gender or racial biases do not think about them consciously or perhaps even realize they have them. I also contend that if we were to dispel the idea that some people are “naturally” gifted and instead recognize that everyone is on a growth journey and can achieve amazing things, some of the most insidious biases against women and people of color would disappear. This is needed in the STEM fields more than anywhere else; it is no coincidence that STEM subjects evidence the strongest fixed thinking and the starkest inequities in participation. 

Part of the reason so many students are dissuaded from thinking they are capable of learning math is the attitudes of the teachers and professors who teach them.

Types of Mistakes

by Eduardo Briceño, community.mindsetworks.com

We can deepen our own and our students' understanding of mistakes, which are not all created equal, and are not always desirable. After all, our ability to manage and learn from mistakes is not fixed. We can improve it.

Here are two quotes about mistakes that I like and use, but that can also lead to confusion if we don't further clarify what we mean:

"A life spent making mistakes is not only most honorable but more useful than a life spent doing nothing"  - George Bernard Shaw

"It is well to cultivate a friendly feeling towards error, to treat it as a companion inseparable from our lives, as something having a purpose which it truly has."  - Maria Montessori

These constructive quotes communicate that mistakes are desirable, which is a positive message and part of what we want students to learn. An appreciation of mistakes helps us overcome our fear of making them, enabling us to take risks. But we also want students to understand what kinds of mistakes are most useful and how to most learn from them.

Types of mistakes

The stretch mistakes

Stretch mistakes happen when we're working to expand our current abilities. We're not trying to make these mistakes in that we're not trying to do something incorrectly, but instead, we're trying to do something that is beyond what we already can do without help, so we're bound to make some errors.

Stretch mistakes are positive. If we never made stretch mistakes, it would mean that we never truly challenged ourselves to learn new knowledge or skills.

Sometimes when we're stuck making and repeating the same stretch mistake, the issue may be that we're mindlessly going through the motions, rather than truly focusing on improving our abilities. Other times the root cause may be that our approach to learning is ineffective and we should try a different strategy to learn that new skill. Or it may be that what we're trying is too far beyond what we already know, and we're not yet ready to master that level of challenge. It is not a problem to test our boundaries and rate of growth, exploring how far and quickly we can progress. But if we feel stuck, one thing we can do is adjust the task, decreasing the level of challenge but still keeping it beyond what we already know. Our zone of proximal development (ZPD) is the zone slightly beyond what we already can do without help, which is a fruitful level of challenge for learning.

We want to make stretch mistakes! We want to do so not by trying to do things incorrectly, but by trying to do things that are challenging. When we make stretch mistakes we want to reflect, identify what we can learn, and then adjust our approach to practice, until we master the new level of ability. Then we want to identify a new area of challenge and continue stretching ourselves.

The aha-moment mistakes

Another positive type of mistake, but one that is harder to strive or plan for, is the aha-moment mistake. This happens when we achieve what we intend to do, but then realize that it was a mistake to do so because of some knowledge we lacked which is now becoming apparent. There are lots of examples of this, such as:

● When we lack the content knowledge: e.g. not finding water, we try to extinguish a fire with alcohol, which we didn't realize is flammable.

● When we find there is more nuance than we realized: e.g. in our painting, we color a sun near the horizon as yellow, and later notice that the sun does not always look yellow.

● When we make incorrect assumptions: e.g. we try to help someone else, thinking that help is always welcome, but we find out that the person did not want help at that moment.

● When we make systematic mistakes: e.g. a fellow educator observes us doing a lesson and later points out, with compelling back-up data, that we tend to call on Caucasian girls much more often than we do other students.

● When we misremember: e.g. we call a friend for their birthday on the right date, but the wrong month.

We can gain more aha moments from mistakes by being reflective. We can ask ourselves What was unexpected? Why did that result occur? What went well and what didn't? Is there anything I could try differently next time? We can also ask people around us for information we may not be aware of, or for ideas for improvement.

The sloppy mistakes

Sloppy mistakes happen when we're doing something we already know how to do, but we do it incorrectly because we lose concentration. We all make sloppy mistakes occasionally because we're human. However, when we make too many of these mistakes, especially on a task that we intend to focus on at the time, it signals an opportunity to enhance our focus, processes, environment, or habits.

Sometimes sloppy mistakes can be turned into aha moments. If we make a mistake because we're not focused on the task at hand, or we're too tired, or something distracted us, upon reflection we can gain aha-moments on how to improve, such as realizing we're better at certain tasks after a good night's sleep, or that if we silence our gadgets or close our doors we can focus better.

The high-stakes mistakes

Sometimes we don't want to make a mistake because it would be catastrophic. For example, in potentially dangerous situations we want to be safe. A big mistake from the person in charge of security in a nuclear power plant could lead to a nuclear disaster. We don't want a school bus driver to take a risk going too fast making a turn, or a student in that bus to blindfold the bus driver. In those cases, we want to put processes in place to minimize high-stakes mistakes. We also want to be clear with students about why we don't want the risk-taking behavior and experimentation in these situations, and how they're different from learning-oriented tasks.

Aside from life-threatening situations, we can sometimes consider performance situations to be high-stakes. For example, if going to a prestigious college is important to someone, taking the SAT could be a high-stakes event because the performance in that assessment has important ramifications. Or if a sports team has trained for years, working very hard to maximize growth, a championship final can be considered a high-stakes event. It is okay to see these events as performance events rather than as learning events, and to seek to minimize mistakes and maximize performance in these events. We're putting our best foot forward, trying to perform as best as we can. How we do in these events gives us information about how effective we have become through our hard work and effort. Of course, it is also ok to embed learning activities in high-stakes events that don't involve safety concerns. We can try something that is beyond what we already know and see how it works, as long as we realize that it may impact our performance (positively or negatively). And of course, we can always learn from these performance events by afterwards reflecting and discussing how things went, what we could do differently next time, and how we could adjust our practice.

In a high-stakes event, if we don't achieve our goal of a high test score or winning the championship, let's reflect on the progress we've made through time, on the approaches that have and haven't helped us grow, and on what we can do to grow more effectively. Then let's go back to spending most of our time practicing, challenging ourselves, and seeking stretch mistakes and learning from those mistakes. On the other hand, if we achieve our target score or win a championship, that's great. Let's celebrate the achievement and how much progress we've made. Then let's ask ourselves the same questions. Let's go back to spending most of our time practicing, challenging ourselves, and growing our abilities.

We're all fortunate to be able to enjoy growth and learning throughout life, no matter what our current level of ability is. Nobody can ever take that source of fulfillment away from us.

Let's be clear

Mistakes are not all created equal, and they are not always desirable. In addition, learning from mistakes is not all automatic. In order to learn from them the most we need to reflect on our errors and extract lessons from them.

If we're more precise in our own understanding of mistakes and in our communication with students, it will increase their understanding, buy-in, and efficacy as learners.

Utangaç Çocuklar

Utangaç bir çocuğun ebeveyni çocuğunun geleceği konusunda genellikle endişelidir. Bilim insanları da bu endişelerin yersiz olmadığını söylüyor, çünkü araştırmalar, çocuklardaki çekingenliğin ileri yaşlarda kaygı bozukluğuna dönüşme riski taşıdığını gösteriyor. Ebeveynlerin utangaç çocuklarını koruma çabalarının ise durumu daha da kötüleştirmesi de mümkün.

Psikologlar ve çocuk gelişimi uzmanları, utangaç çocukları desteklemenin yollarını arıyor. New York Üniversitesi’nden psikolog Sandee McClowry’ye göre yapılması gereken şey, çocukların temel yapısal özelliklerini değiştirmeye çalışmadan, onları kendilerini rahat hissettikleri bölgelerin dışına çıkmaya ikna etmek. Onları oldukları gibi kabul etmek, utangaç çocuklar için çok önemli.

Psikologlar utangaçlığı sosyallikten kaçınma, sosyal etkileşimlere maruz kalınması durumunda ise sıkıntı ve gerginlik hissedilmesi olarak tanımlamaktadır. Utangaçlık üzerine çalışan araştırmacılar,  hem insanlarla tanıştıklarında, hem de ilk defa karşılaştıkları durumlarda kaygıları tetiklenen çocukları daha iyi teşhis etmek için daha geniş bir kavram olan davranış tutukluluğundan yararlanıyor.

Utangaçlık, çocuklarda bir karakter özelliğidir. Psikologlar bu gibi karakteristik özelliklerin oldukça ısrarcı olduğunu belirtiyor. 1988 yılında Child Development dergisinde yayınlanan bir araştırmada 4 yaşında çocukların davranışları incelenmiş, aynı çocuklar 7 buçuk yaşındayken bir inceleme daha yapılmış. Araştırma sonucunda 4 yaşındayken utangaç olan çocukların da girişken olan çocukların da 7 buçuk yaşına geldiklerinde yine aynı davranışları sergiledikleri gözlemlenmişti.

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Aşırı korumacı ebeveynlerin aslında onlara zarar verdiğini belirten McClowry, anne ve babaların davranışlarında belirli bir denge tutturması gerektiğini vurguluyor. “Yapı iskelesi” olarak adlandırılan teknik, utangaç çocukların ebeveynleri için oldukça uygun görünüyor. Eğitimde kullanılan “yapı iskelesi” tekniği, öğrencilere başta yoğun destek verip bu desteği yavaş yavaş ve düzenli olarak azaltarak onların daha bağımsız hale gelmesini sağlamak anlamına geliyor. Bu teknik, utangaç çocukların kabuklarından çıkmalarına yardımcı olabilir.

McClowry, bu teknikle ilgili olarak kamp örneğini veriyor. Örneğin bir çocuk kampa gitmek istiyor ancak geceyi evden uzakta geçirmekten korkuyorsa, anne ve baba işe, çocuğun arkadaşlarını evlerinde kalmaya davet etmekle başlayabilir. Ardından bir geceyi büyükannelerinin evinde geçirerek çıtayı yavaş yavaş yükseltebilirler. Elbette anne ve baba “yapı iskelesi” tekniğini uygularken çocuğun rahatsızlık hissedip hissetmediğini de kontrol etmeli, çocuk daha fazla katlanamaz hale gelirse onu zorlamamalıdır. Ayrıca çocukların daha büyük olduğu durumlarda ebeveynler bu tecrübeyi onunla konuşabilir, ona nasıl hissettiğini, neyin daha iyi hissetmesini sağladığını ve bu adımdan sonra ne yapmak istediğini sorabilir.

Kaynak: Herkese Bilim Teknoloji

Middle School Misfortunes Then and Now

By: Benjamin Conlon, waituntil8th

Let’s imagine a seventh grader. He’s a quiet kid, polite, with a few friends. Just your ordinary, run-of-the-mill twelve-year-old. We’ll call him Brian. Brian’s halfway through seventh grade and for the first time, he’s starting to wonder where he falls in the social hierarchy at school. He’s thinking about his clothes a little bit, his shoes too. He’s conscious of how others perceive him, but he’s not that conscious of it. 

He goes home each day and from the hours of 3 p.m. to 7 a.m., he has a break from the social pressures of middle school. Most evenings, he doesn’t have a care in the world. The year is 2008. 

Brian has a cell phone, but it’s off most of the time. After all, it doesn’t do much. If friends want to get in touch, they call the house. The only time large groups of seventh graders come together is at school dances. If Brian feels uncomfortable with that, he can skip the dance. He can talk to teachers about day-to-day problems. Teachers have pretty good control over what happens at school.

Now, let’s imagine Brian on a typical weekday. He goes downstairs and has breakfast with his family. His mom is already at work, but his dad and sisters are there. They talk to each other over bowls of cereal. The kids head off to school soon after. Brian has a fine morning in his seventh grade classroom and walks down to the lunchroom at precisely 12 p.m.

There’s a slick of water on the tiled floor near the fountain at the back of the cafeteria. A few eighth graders know about it, and they’re laughing as yet another student slips and tumbles to the ground.

Brian buys a grilled cheese sandwich. It comes with tomato soup that no one ever eats. He polishes off the sandwich and heads to the nearest trashcan to dump the soup. When his sneakers hit the water slick, he slips just like the others. The tomato soup goes up in the air and comes down on his lap. 

Nearby, at the table of eighth graders, a boy named Mark laughs. He laughs at Brian the same way the boys around him laugh at Brian. They laugh because they’re older, and they know something the younger kids don’t. They laugh at the slapstick nature of the fall. The spilled tomato soup is a bonus. The fall is a misfortune for Brian. That’s all. It’s not an asset for Mark. A few kids hear the laughter and look over, but Brian gets up quickly and rushes off to the bathroom to change into his gym shorts.

Mark tries to retell the story to a friend later. The friend doesn’t really get it because he wasn’t there. He can’t picture it. In fact, Mark seems a little mean for laughing at all.

After lunch, Brian returns to homeroom in his gym shorts. No one seems to notice the change. He breathes a sigh of relief. The cafeteria fall is behind him. He meets his sisters at the end of the day and they ask why he’s wearing gym shorts. He tells them he spilled some tomato sauce on his pants. They head home and spend the afternoon and evening together, safe and sound, home life completely separate from school life. Brian doesn’t think about the incident again. Only a few people saw it. It’s over. 

Now, let’s imagine Brian again. Same kid. Same family. Same school. He’s still in seventh grade, but this time it’s 2018. 

When Brian sits down for breakfast, his dad is answering an email at the table. His older sister is texting, and his younger sister is playing a video game. Brian has an iPhone too. He takes it out and opens the Instagram app. The Brian from 2008 was wondering about his position in the social hierarchy. The Brian from 2018 knows. He can see it right there on the screen. He has fewer ‘followers’ than the other kids in his grade. That’s a problem. He wants to ask his father what to do, but there’s that email to be written. Instead, Brian thinks about it all morning at school. While his teacher talks, he slips his phone out and checks to see how many ‘followers’ the other kids in class have. The answer doesn’t help his confidence. At precisely 12 p.m., he heads to the cafeteria. He buys a grilled cheese. It comes with tomato soup that no one ever eats. 

At the back of the lunchroom, Mark sits with the other eighth graders. He holds a shiny new iPhone in one hand. Mark has had an iPhone for five years. He’s got all the apps. Twitter, Instagram and Snapchat. He’s got lots of followers too. He doesn’t know all of them, but that’s okay.

A few years ago, Mark made his first Instagram post. It was a picture of his remote control car. Mark used to really enjoy remote control cars. Mark checked Instagram an hour after putting up that first picture. A bright red dot showed at the bottom of the page. He clicked it. Someone had ‘liked’ the picture of the car. Mark felt validated. It was good that he posted the picture. A little bit of dopamine was released into Mark’s brain. He checked the picture an hour later. Sure enough, another ‘like’. More dopamine. He felt even better. 

For a while, pictures of the remote control car were sufficient. They generated enough ‘likes’ to keep Mark happy. He no longer got much joy from actually driving the remote control car, but he got plenty from seeing those ‘likes’ pile up. 

Then something started to happen. The ‘likes’ stopped coming in. People didn’t seem interested in the pictures of the car anymore. This made Mark unhappy. He missed the ‘likes’ and the dopamine that came with them. He needed them back. He needed more exciting pictures, because exciting pictures would bring more views and more ‘likes’. So, he decided to drive his car right out into the middle of the road. He had his little brother film the whole thing. He filmed the remote control car as it got flattened by a passing truck. Mark didn’t bother to collect it. He just grabbed his phone and posted the video. It was only a few minutes before the ‘likes’ started coming in. He felt better. 

Now it’s eighth grade and Mark has become addicted to social media.  Sure, he needs a lot more ‘likes’ to get the same feeling, but that’s okay. That just means he needs more content. Good content. Content no one else has. That’s the kind that gets a lot of ‘likes’, really, really fast. Mark has learned the best content comes from filming and posting the embarrassing experiences of classmates. 

When he notices that water slick at the back of the cafeteria, he’s ready.  Each time someone walks by and falls, their misfortune becomes an asset for Mark. A part of Mark wants them to fall. He hopes they fall.

Brian walks across the cafeteria with his soup, minding his own business. Suddenly, his feet slide out from under him. The tomato soup goes up in the air and comes down on his lap. He’s so embarrassed, that when he stands up and rushes off to the bathroom, he doesn’t notice Mark filming.

Mark’s fingers race over his iPhone screen before Brian is out of sight. That was a great video he just took, and he wants to get it online. Fast. He knows he’s not supposed to have his cell phone out in school, but the teachers really only enforce that rule during class. They all use Twitter and Instagram too. They understand. 

Mark doesn’t know who he just filmed, and he doesn’t care. It’s not his fault the kid fell on the floor. He’s just the messenger. The video is a kind of public service announcement. He’s just warning everyone else about the water spot in the cafeteria. That’s what Mark tells himself.

He gets the video uploaded to Snapchat first. No time for a caption. It speaks for itself. He has it up on Instagram seconds later. By then, the ‘likes’ are already coming in. Dopamine floods into Mark’s brain. There’s a comment on Instagram already! “What a loser!” it says. Mark gives the comment a ‘like’. Best to keep the audience happy. 

This has been a rewarding lunch. The bell’s going to ring in a few minutes. Mark sits back and refreshes his screen again and again and again until it does.

Meanwhile, Brian heads back from the bathroom, having changed into his gym shorts. He’s still embarrassed about the fall. It happened near the back of the cafeteria, though. He doesn’t think many people saw. He hopes they didn’t. But when he walks into the classroom, a lot of people look at him. One girl holds her phone up at an odd angle. Is she…taking a picture? The phone comes down quickly and she starts typing, so he can’t be sure. 

Class begins. Brian is confused because people keep slipping their phones out and glancing back at him. He asks to go to the bathroom. Inside a stall, he opens Instagram. There he is on the screen, covered in tomato sauce. How could this be? Who filmed this? Below the video, a new picture has just appeared. It’s him in his gym shorts. The caption reads, “Outfit change!”

Brian scrolls frantically through the feed trying to find the source of the video. He can’t. It’s been shared and reshared too many times. He notices his follower count has dropped. He doesn’t want to go to class. He just wants it to stop. 

He meets his sisters outside at the end of the day. Several students snap pictures as he walks by. Neither sister says a word. Brian knows why. 

Home was a safe place for Brian in 2008. Whatever happened in school, stayed in school. Not now. Brian arrives at his house, heart thundering, and heads straight to his bedroom. He’s supposed to be doing homework, but he can’t concentrate. Alone in the dark, he refreshes his iPhone again and again and again and again.

Brian’s family is having his favorite dish for dinner, but he doesn’t care. He wants it to be over so he can get back to his phone. Twice, he goes to the bathroom to check Instagram. His parents don’t mind, they’re checking their own phones.

Brian discovers that two new versions of the video have been released. One is set to music and the other has a nasty narration. Both have lots of comments. He doesn’t know how to fight back, so he just watches as the view counts rise higher and higher. His own follower count, his friend count, keeps going in the opposite direction. Brian doesn’t want to be part of this. He doesn’t like this kind of thing. He can’t skip it though. It’s not like the dance. And he can’t tell a teacher. This isn’t happening at school.

He stays up all night refreshing the feed, hoping the rising view count will start to slow. Mark is doing the same thing at the other side of town. He has lots of new followers. This is his best video ever. 

At 3 a.m., they both turn off their lights and stare up at their respective ceilings. Mark smiles. He hopes tomorrow something even more embarrassing happens to a different kid. Then he can film that and get even more ‘likes’. Across town, Brian isn’t smiling, but sadly, he’s hoping for exactly the same thing. 

From the Author

I started teaching in 2009. At that time, public school was very much the way I remembered it. That’s not the case anymore. Smartphones and social media have transformed students into creatures craving one thing: content. It’s a sad state of affairs. 

But there’s hope. 

Over the last few years, my students have become increasingly interested in stories from the days before smartphones and social media. In the same way many adults look back fondly on simpler times, kids look back to second and third grade, when no one had a phone. I think a lot of them already miss those days. 

Smartphones and social media aren’t going anywhere. Both are powerful tools, with many benefits. But they have fundamentally altered how children interact with the world and not in a good way. We can change that. In addition to the “Wait Until 8th” pledge, consider taking the following steps to help your children reclaim childhood.

1. Propose that administrators and teachers stop using social media for school related purposes. In many districts teachers are encouraged to employ Twitter and Instagram for classroom updates. This is a bad thing. It normalizes the process of posting content without consent and teaches children that everything exciting is best viewed through a recording iPhone. It also reinforces the notion that ‘likes’ determine value. Rather than reading tweets from your child’s teacher, talk to your children each day. Ask what’s going on in school. They’ll appreciate it.

2. Insist that technology education include a unit on phone etiquette, the dark sides of social media and the long-term ramifications of posting online. Make sure students hear from individuals who have unwittingly and unwillingly been turned into viral videos.   

3. Tell your children stories from your own childhood. Point out how few of them could have happened if smartphones had been around. Remind your children that they will some day grow up and want stories of their own. An afternoon spent online doesn’t make for very good one.

4. Teach your children that boredom is important. They should be bored. Leonardo Da Vinci was bored. So was Einstein. Boredom breeds creativity and new ideas and experiences. Cherish boredom. 

5. Remind them that, as the saying goes, adventures don’t come calling like unexpected cousins. They have to be found. Tell them to go outside and explore the real world. Childhood is fleeting. It shouldn’t be spent staring at a screen.

Mathematical Mindsets [Jo Boaler] (4): What is mathematics?

Jenya Sapir

When we ask students what math is, they will typically give descriptions that are very different from those given by experts in the field. Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what math is, they will say it is the study of patterns; that it is an aesthetic, creative, and beautiful subject (Devlin, 1997). Why are these descriptions so different?

**

Recently I was chairing the PhD viva of one of Maryam's students. A viva is the culminating exam for PhD students when they “defend” the dissertations they have produced over a number of years in front of their committee of professors. I walked into the math department at Stanford that day, curious about the defense I was to chair. The room in which the defense was held was small, with windows overlooking Stanford's impressive Palm Drive, the entrance to the university, and it was filled with mathematicians, students, and professors who had come to watch or judge the defense. Maryam's student was a young woman names Jenya Sapir, who strode up and down that day, sharing drawings on different walls of the room, pointing to them as she made conjectures about the relationships between lines and curves on her drawings. The mathematics she described was a subject of visual images, creativity, and connections, and it was filled with uncertainty

**

Three or four times in the defense, professors asked questions, to which the confident young woman simply answered, “I don't know.” Often the professor added that she or he did not know either. It would be very unusual in a defense of an education PhD for a student to give the answer: “I don't know,” and it would be frowned upon by some professors. But mathematics, real mathematics, is a subject full of uncertainty; it is about explorations, conjectures, and interpretations, not definitive answers. The professors thought it was perfectly reasonable that she did not know the answers to some of the questions, as her work was entering uncharted territories. She passed the PhD exam with flying colors.

This does not mean that there are no answers in mathematics. Many things are known and are important for students to learn. But somehow school mathematics has become so far removed from real mathematics that if I had taken most school students into the mathematics department defense that day, they would not have recognized the subject before them. This wide gulf between real mathematics and school mathematics is at the heart of the math problems we face in education. I strongly believe that if school math classrooms presented the true nature of the discipline, we would not have this nationwide dislike of math and widespread math underachievement.

Mathematics is a cultural phenomenon; a set of ideas, connections, and relationships that we can use to make sense of the world. At its core, mathematics is about patterns. We can lay a mathematical lens upon the world, and when we do, we see patterns everywhere; and it is through our understanding of the patterns, developed through mathematical study, that new and powerful knowledge is created.

**

Knowledge of mathematical patterns has helped people navigate oceans, chart missions to space, develop technology that powers cell phones and social networks, and create new scientific and medical knowledge, yet many school students believe that math is a dead subject, irrelevant to their futures.

To understand the real nature of mathematics it is helpful to consider the mathematics in the world—the mathematics of nature. The patterns that thread through oceans and wildlife, structures and rainfall, animal behavior, and social networks have fascinated mathematicians for centuries.and social networks have fascinated mathematicians for centuries.

**

Numerous research studies (Silver, 1994) have shown that when students are given opportunities to pose mathematics problems, to consider a situation and think of a mathematics question to ask of it—which is the essence of real mathematics—they become more deeply engaged and perform at higher levels. But this rarely happens in mathematics classrooms. In A Beautiful Mind, the box office movie hit, viewers watch John Nash (played by Russell Crowe) strive to find an interesting question to ask—the critical and first stage of mathematical work. In classrooms students do not experience this important mathematical step; instead, they spend their time answering questions that seem dead to them, questions they have not asked.

**

Over the years, school mathematics has become more and more disconnected from the mathematics that mathematicians use and the mathematics of life. Students spend thousands of hours in classrooms learning sets of procedures and rules that they will never use, in their lives or in their work. Conrad Wolfram is a director of Wolfram-Alpha, one of the most important mathematical companies in the world. He is also an outspoken critic of traditional mathematics teaching, and he argues strongly that mathematics does not equal calculating. In a TED talk watched by over a million people, Wolfram (2010) proposes that working on mathematics has four stages:

1. Posing a question

2. Going from the real world to a mathematical model

3. Performing a calculation

4. Going from the model back to the real world, to see if the original question was answered

The first stage involves asking a good question of some data or a situation—the first mathematical act that is needed in the workplace. The fastest-growing job in the United States is that of data analyst—someone who looks at the “big data” that all companies now have and asks important questions of the data. The second stage Wolfram describes is setting up a model to answer the  question; the third is performing a calculation, and the fourth is turning the model back to the world to see whether the question is answered. Wolfram points out that 80% of school mathematics is spent on stage 3—performing a calculation by hand—when that is the one stage that employers do not need workers to be able to do, as it is performed by a calculator or computer. Instead, Wolfram proposes that we have students working on stages 1, 2, and 4 for much more of their time in mathematics classes.

What employers need, he argues, is people who can ask good questions, set up models, analyze results, and interpret mathematical answers. It used to be that employers needed people to calculate; they no longer need this. What they need is people to think and reason.

The Fortune 500 comprises the top 500 companies in the United States. Forty-five years ago, when companies were asked what they most valued in new employees, the list looked like this:

Computation has dropped to the second-from-the-last position, and the top places have been taken by teamwork and problem solving.

Parents often do not see the need for something that is at the heart of mathematics: the discipline. Many parents have asked me: What is the point of my child explaining their work if they can get the answer right? My answer is always the same: Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics. Scientists prove or disprove theories by producing more cases that do or do not work, but mathematicians prove theories through mathematical reasoning. They need to produce arguments that convince other mathematicians by carefully reasoning their way from one idea to another, using logical connections. Mathematics is a very social subject, as proof comes about when mathematicians can convince other mathematicians of logical connections.

A lot of mathematics is produced through collaborations between mathematicians; Leone Burton studied the work of mathematicians and found that over half of their publications were produced collaboratively (Burton, 1999). Yet many mathematics classrooms are places where students complete worksheets in silence. Group and whole class discussions are really important. Not only are they the greatest aid to understanding—as students rarely understand ideas without talking through them—and not only do they enliven the subject and engage students, but they teach students to reason and to critique each other's reasoning, both of which are central in today's high-tech workplaces. Almost all new jobs in today's technological world involve working with massive data sets, asking questions of the data and reasoning about pathways. Conrad Wolfram told me that anyone who cannot reason about mathematics is ineffective in today's workplace. When employees reason and talk about mathematical pathways, other people can develop new ideas based on the pathways as well as see if a mistake has been made. The teamwork that employers value so highly is based upon mathematical reasoning. People who just give answers to calculations are not useful in the workplace; they must be able to reason through them.

Mathematical Mindsets [Jo Boaler] (5): Real Mathematicians are Slow Thinkers!

Mathematical Mindsets [Jo Boaler] (3):The Power of Mistakes and Struggle

Mathematical Mindsets [Jo Boaler] (2):The Brain and Mathematics Learning

Mathematical Mindsets [Jo Boaler] (1):Traumatized by Math